Applied Mathematical Sciences, Vol. 8, 2014, no. 67, 3321-3330 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.44265 Homomorphism and Cartesian Product of Fuzzy PS Algebras T. Priya Department of Mathematics PSNA College of Engineering and Technology Dindigul 624 622, Tamilnadu, India T. Ramachandran Department of Mathematics M.V.Muthiah Government Arts College for Women Dindigul 624 001, Tamilnadu, India Copyright 2014 T. Priya and T. Ramachandran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce the concept of fuzzy PS-ideal of PS-algebra under homomorphism and some of its properties. We proved that is a fuzzy PS-ideal(PSsubalgebra) of a PS-algebra X iff is a fuzzy PS-ideal(PS-subalgebra) of X x X, where is the strongest fuzzy relation. Mathematics Subject Classification: 20N25, 03E72, 03F055, 06F35, 03G25 Keywords: PS-algebra, fuzzy PS-subalgebra, fuzzy PS- ideal, homomorphism, Cartesian product
3322 T. Priya and T. Ramachandran 1. Introduction K.Iseki and S.Tanaka[1,2] introduced two classes of abstract algebras : BCK-algebras and BCI algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. J.Neggers, S.S.Ahn and H.S.Kim[3] introduced Q-algebras and d-algebras which is generalization of BCK / BCI algebras and obtained several results. C.Prabpayak and U.Leerawat [4] introduced a new algebraic structure which is called KU-algebras and investigated some properties. The concept of fuzzy set was introduced by L.A.Zadeh in 1965 [14]. Since then these ideas have been applied to other algebraic structures such as groups, rings, modules, vector spaces and topologies. T.Priya and T.Ramachandran [5,6,7,8] introduced the new algebraic structure, PS-algebra, which is an another generalization of BCI / BCK/Q /d/ KU algebras and investigated its properties related to fuzzy, fuzzy dot in detail. In this paper, We investigate the behavior of fuzzy PS-ideal and PS-subalgebra of PS-algebra, with the homomorphism and Cartesian products and obtain some of its results. 2. Preliminaries In this section we site the fundamental definitions that will be used in the sequel. Definition 2.1 [5] A nonempty set X with a constant 0 and a binary operation * is called PS Algebra if it satisfies the following axioms. 1. x * x = 0 2. x * 0 = 0 3. x * y = 0 and y * x = 0 x = y, x,y X. Example 2.1 Let X = { 0,1,2 } be the set with the following table. * 0 1 2 0 0 1 2 1 0 0 1 2 0 2 0 Then (X, *, 0 ) is a PS Algebra.
Homomorphism and cartesian product 3323 Definition 2.2 [5,6] Let X be a PS-algebra and I be a subset of X, then I is called a PS-ideal of X if it satisfies the following conditions: 1. 0 I 2. y * x I and y I x I Definition 2.3 [14] Let X be a non-empty set. A fuzzy subset of the set X is a mapping : X [0,1]. Definition 2.4 [6,7] Let X be a PS-algebra. A fuzzy set in X is called a fuzzy PS-ideal of X if it satisfies the following conditions. (i) (0) (x) (ii) (x ) min{ (y *x), (y)},for all x,y X Definition 2.5 [6,8,9] y)}, for all x,y X. A fuzzy set in a PS-algebra X is called a fuzzy PS-subalgebra of X if (x * y) min { (x ), ( 3. Homomorphism on Fuzzy PS-algebras In this section, we discussed about Fuzzy PS-ideals and PS-subalgebra in PS-algebra under homomorphism and some of its properties. Definition 3.1 [8,12,13] Let (X,*,0) and ( Y,*,0` ) be PS algebras. A mapping f: X Y is said to be a homomorphism if f( x * y) = f(x) * f(y) for all x,y X. Remark: If f: X Y is a homomorphism of PS-algebra, then f(0) = 0. Definition 3.2 [8,12] Let f: X X be an endomorphism and µ be a fuzzy set in X. We define a new fuzzy set in X by µ f in X as µ f (x) = µ (f(x)) for all x in X. Theorem 3.3 µ f. Let f be an endomorphism of a PS- algebra X. If µ is a fuzzy PS-ideal of X, then so is
3324 T. Priya and T. Ramachandran Let µ be a fuzzy PS-ideal of X. Now, µ f (x) = µ (f(x)) µ (f(0)) = µ f (0), for all x X. µ f (0) µ f (x) Let x,y X Then µ f ( x ) = µ ( f( x)) min { µ (f(y) * f(x)), µ(f (y))} = min { µ ( f( y * x ) ), µ(f (y))} = min { µ f ( y * x ), µ f (y)} µ f ( x ) min { µ f ( y * x ), µ f (y)} Hence µ f is a fuzzy PS-ideal of X. Theorem 3.4 Let f: X Y be an epimorphism of PS- algebra. If µ f is a fuzzy PS-ideal of X, then µ is a fuzzy PS-ideal of Y. Let µ f be a fuzzy PS-ideal of X and let y Y. Then there exists x X such that f(x) = y. Now, µ (0) = µ (f(0)) = µ f (0) µ f (x) = µ (f(x)) = µ (y) (0) µ (y ) Let y 1, y 2 Y. µ (y 1 ) = µ (f (x 1 )) = µ f ( x 1 ) min { µ f ( x 2 * x 1 ), µ f ( x 2 )} = min {µ (f (x 2 * x 1 )), µ(f(x 2 ))} = min {µ (f (x 2 ) * f(x 1 )), µ (f( x 2 ))} = min {µ (y 2 * y 1 ), µ(y 2 )} µ (y 1 ) min { µ ( y 2 * y 1 ), µ(y 2 )} µ is a fuzzy PS-ideal of Y. Theorem 3.5 Let f: X Y be a homomorphism of PS- algebra. If µ is a fuzzy PS-ideal of Y then µ f is a fuzzy PS-ideal of X. Let µ be a fuzzy PS-ideal of Y and let x,y X. Then f (0) = (f(0)) (f(x)) = f (x) f (0) f (x). Also µ f (x) = µ ( f (x ) )
Homomorphism and cartesian product 3325 min { µ ( f(y) * f(x) ), µ ( f (y) ) } = min { µ ( f( y * x) ), µ ( f (y) ) } = min { µ f ( y * x), µ f (y) } µ f ( x ) min { µ f ( y * x), µ f (y) }. Hence µ f is a fuzzy PS-ideal of X. Theorem 3.6 Let f : X Y be a homomorphism of a PS-algebra X into a PS-algebra Y. If µ is a fuzzy PS- subalgebra of Y, then the pre- image of µ denoted by f -1 (µ), defined as {f -1 (µ)}(x) = µ(f(x)), x X, is a fuzzy PS- subalgebra of X. Let µ be a fuzzy PS- subalgebra of Y. Let x, y X. Now, {f -1 (µ)}(x* y) = ( f (x *y) ) = ( f (x) * f(y) ) min { (f (x)), (f(y)) } = min {{f -1 (µ)} (x), {f -1 (µ)}(y) } f -1 (µ) is a fuzzy PS-subalgebra of X. Theorem 3.7 If µ be a fuzzy PS- subalgebra of X, then f is also a fuzzy PS-subalgebra of X. Let µ f be a fuzzy PS- subalgebra of X. Let x, y X. Now, µ f (x* y) = ( f (x *y) ) = ( f (x) * f(y) ) min { (f (x)), (f(y)) } = min {µ f (x), µ f (y)} µ f is a fuzzy PS-subalgebra of X. 4. Cartesian Product of Fuzzy PS-ideals of PS algebras In this section, we discuss the concept of Cartesian product of fuzzy PS-ideal and PSsubalgebra of PS-algebra. Definition 4.1 [8] Let µ and be the fuzzy sets in X. The Cartesian product µ x : X x X [0,1] is defined by ( µ x ) ( x, y) = min { (x), (y)}, for all x, y X.
3326 T. Priya and T. Ramachandran Theorem 4.2 If µ and are fuzzy PS-ideals in a PS algebra X, then µ x is a fuzzy PS-ideal in X x X. Let ( x 1, x 2 ) X x X. (µ x ) (0,0) = min { µ (0), (0) } min { µ (x 1 ), ( x 2 ) } = ( µ x ) (x 1, x 2 ) Let ( x 1, x 2 ), ( y 1, y 2 ) X x X. Now, ( µ x ) ( x 1, x 2 ) = min { µ ( x 1 ), ( x 2 )} min {min {µ(y 1 * x 1 ), µ( y 1 )},min { ( y 2 * x 2 ), ( y 2 )}} = min {min {µ(y 1 * x 1 ), ( y 2 * x 2 )}, min { µ( y 1 ), (y 2 )}} = min {( µ x ) ( (y 1, y 2 ) * (x 1, x 2 ) ), (µ x ) (y 1, y 2 )} (µx ) (x 1, x 2 ) min {( µ x ) ( (y 1, y 2 ) * (x 1, x 2 ) ), (µ x ) (y 1, y 2 )}. Hence, µ x is a fuzzy PS- ideal in X x X. Theorem 4.3: Let & be fuzzy sets in a PS-algebra X such that x is a fuzzy PS-ideal of X x X. Then (i) Either (0) (x) (or) (0) (x) for all x X. (ii) If (0) (x) for all x X, then either (0) (x) (or) (0) (x) (iii) If (0) (x) for all x X, then either (0) (x) (or) (0) (x) Let x be a fuzzy PS-ideal of X x X. (i) Suppose that (0) < (x) and (0) < (x) for some x, y X. Then ( x ) (x,y) = min{ (x), (y) } > min { (0), (0) } = ( x ) (0,0), Which is a contradiction. Therefore (0) (x) or (0) (x) for all x X. (ii) Assume that there exists x,y X such that (0) < (x) and (0) < (x). Then ( x ) (0,0) = min { (0), (0) } = (0) and hence ( x ) (x, y) = min { (x), (y) } > (0) = ( x ) (0,0) Which is a contradiction. Hence, if (0) (x) for all x X, then either (0) (x) (or) (0) (x).
Homomorphism and cartesian product 3327 Similarly, we can prove that if (0) (x) for all x X, then either (0) (x) (or) (0) (x), which yields (iii). Theorem 4.4 Let & be fuzzy sets in a PS-algebra X such that x is a fuzzy PS-ideal of X x X. Then either or is a fuzzy PS-ideal of X. First we prove that is a fuzzy PS-ideal of X. Since by 4.3(i) either (0) (x) (or) (0) (x) for all x X. Assume that (0) (x) for all x X. It follows from 4.3(iii) that either (0) (x) (or) (0) (x). If (0) (x), for any x X, then (x) = min { (0), (x)}= ( x ) ((0, x) (x) = ( x ) (0, x) min {(µ x )((0,y)* (0,x)), ( x ) (0, y)} = min {(µ x ) ( (0*0),(y*x) ), ( x ) (0, y)} = min {(µ x ) (0,(y*x) ), ( x ) (0, y)} = min { (y*x), (y)} Hence is a fuzzy PS-ideal of X. Next we will prove that is a fuzzy PS-ideal of X. Let (0) (x).since by theorem 4.3(ii), either (0) (x) (or) (0) (x). Assume that (0) (x), then (x) = min { (x), (0)} = ( x ) (x,0) ( x ) = ( x ) (x,0) min{(µ x )( (y,0) * (x,0) ), ( x ) (y,0)} = min {(µ x ) ( (y * x), (0*0) ), ( x ) (y,0)} = min {(µ x ) ( (y * x), 0 ), ( x ) (y,0)} = min { (y * x), (y)} Hence is a fuzzy PS-ideal of X. Theorem 4.5 If and are fuzzy PS-subalgebras of a PS-algebra X, then x is also a fuzzy PSsubalgebra of X x X.
3328 T. Priya and T. Ramachandran Proof : For any x 1, x 2, y 1, y 2 X. ( x )(( x 1, y 1 ) *( x 2, y 2 )) = ( x )( x 1 * x 2, y 1 * y 2 ) = min { ( x 1 * x 2 ), ( y 1 * y 2 ) } min { min { (x 1 ), (x 2 ) },min{ ( y 1 ), ( y 2 ) } } = min { min { (x 1 ), (y 1 ) }, min { (x 2 ), ( y 2 ) } } = min { ( x ) ( x 1, y 1 ), ( x ) ( x 2, y 2 ) } This completes the proof. Definition 4.6 Let be a fuzzy subset of X. The strongest fuzzy -relation on PS-algebra X is the fuzzy subset of X x X given by (x, y) = min{ (x), (y) }, for all x, y X. Theorem 4.7 Let be the strongest fuzzy -relation on PS-algebra X, where is a fuzzy set of a PS-algebra X. If is a fuzzy PS-ideal of a PS-algebra X, then is a fuzzy PS-ideal of X x X. Proof : Let be a fuzzy PS-ideal of a PS-algebra X. Let (x 1, x 2 ), (y 1, y 2 ) X x X. Then (0,0) = min { (0), (0) } min { (x 1 ), (x 2 ) }= ( x 1, x 2 ) and also ( x 1, x 2 ) = min { (x 1 ), ( x 2 ) } min { min { (y 1 * x 1 ), (y 1 ) }, min { (y 2 * x 2 ), (y 2 ) } } = min {min { (y 1 * x 1 ), (y 2 * x 2 ) }, min { (y 1 ), (y 2 )}} = min { ( (y 1 * x 1 ), (y 2 * x 2 ) ), (y 1, y 2 ) } = min { ( (y 1, y 2 ) * ( x 1, x 2 ) ), (y 1, y 2 ) } Therefore is a fuzzy PS-ideal of X x X. Theorem 4.8 If is a fuzzy PS-ideal of X x X, then is a fuzzy PS-ideal of a PS-algebra X. Proof : Let is a fuzzy PS-ideal of X x X. Then for all (x 1, x 2 ), (y 1, y 2 ) X x X. min { (0), (0) } = (0,0) ( x 1, x 2 ) = min { (x 1 ), (x 2 ) } (0) (x 1 ) or (0) (x 2 ). Also, min { ( x 1 ), ( x 2 ) } = ( x 1, x 2 ) min { ( (y 1, y 2 ) * ( x 1, x 2 ) ), (y 1, y 2 ) } = min { ( (y 1 * x 1 ), (y 2 * x 2 ) ), (y 1, y 2 ) } = min {min { (y 1 * x 1 ), (y 2 * x 2 ) }, min { (y 1 ), (y 2 )}} = min { min { (y 1 * x 1 ), (y 1 ) }, min { (y 2 * x 2 ), (y 2 ) } } Put x 2 = y 2 = 0, we get ( x 1 ) min { (y 1 * x 1 ), (y 1 ) } Hence is a fuzzy PS-ideal of a PS-algebra X.
Homomorphism and cartesian product 3329 Theorem 4.9 If is a fuzzy PS-subalgebra of a PS-algebra X, then is a fuzzy PS-subalgebra of X x X. Proof : Let be a fuzzy PS-subalgebra of a PS-algebra X and let x 1, x 2, y 1, y 2 X. Then (( x 1, y 1 ) *( x 2, y 2 )) = ( x 1 * x 2, y 1 * y 2 ) = min { ( x 1 * x 2 ), ( y 1 * y 2 ) } min { min { (x 1 ), (x 2 ) }, min { (y 1 ), (y 2 ) } } = min { min { (x 1 ), (y 1 ) }, min { (x 2 ), ( y 2 ) } } = min { ( x 1, y 1 ), ( x 2, y 2 ) } Therefore is a fuzzy PS-subalgebra of X x X. Theorem 4.10 If is a fuzzy PS-subalgebra of X x X, then is a fuzzy PS-subalgebra of a PSalgebra X. Proof : Let x, y X. Now, ( x * y ) = min { (x * y), (x * y) } = ((x * y) * (x * y)) min { (x * y), (x * y) } = min { min { (x), (y)}, min { (x), (y)} } = min { (x), (y)} ( x * y ) min { (x), (y) }, which completes the proof. 5. Conclusion In this article authors have been discussed fuzzy PS-ideals and PS-subalgebras in fuzzy PS-algebra under homomorphism and Cartesian product. It has been observed that PSalgebra as a generalization of BCK/BCI/Q/d/TM/KU-algebras. Interestingly, the strongest fuzzy -relation on PS-algebra concept has been discussed in Cartesian products and it adds an another dimension to the defined fuzzy PS--algebra. This concept can further be generalized to Intuitionistic fuzzy set, interval valued fuzzy sets, Anti fuzzy sets for new results in our future work. Acknowledgement Authors wish to thank Dr. K.T.Nagalakshmi,Professor and Head, Department of Mathematics, KLN College of Information and Technology, Pottapalayam, Sivagangai District, Tamilnadu, India.
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